Endomorphisms of hypersurfaces and other manifolds (Q2708375)

Let \(X\) be a smooth complex projective hypersurface of dimension at least 2. In the paper under review, the author studies the problem of the existence of endomorphisms of \(X\) of degree greater than one. In the case of quadrics of dimension at least 3, \textit{K. H. Paranjape} and \textit{V. Srinivas} proved that such endomorphisms do not exist [Invent. Math. 98, No. 2, 425-444 (1989; Zbl 0697.14037)]. NEWLINENEWLINENEWLINEHere the analogous result is proved for hypersurfaces of degree at least 3. Moreover, some general remarks are made on projective smooth varieties \(X\) which admit such an endomorphism \(f\). For example, the Kodaira dimension \(\kappa(X)\) is less than \(\dim X\), and if \(\kappa(X)\geq 0\) then \(f\) is étale. Moreover for a Del Pezzo surface the existence of such an endomorphism is equivalent to the condition \(K_X^2\geq 6\).